To study the concentration of noise averaging \(\bar{}_t\) , we use the fact that \(\lbrace _t\rbrace _{t=0}^\infty \)  is a martingale difference sequence, and rely on concentration inequalities for matrix martingales. These concentration inequalities require a sub-exponential tail condition on the noise. We say that a random variable \(X\)  is \(K\) -sub-exponential if \(\mathbb {E}[|X|^p]\le p!\cdot K^p/2\)  for all \(p=2,3,...\,\) , which is consistent (up to constants) with all standard notions of sub-exponentiality [1]}.
